Found problems: 68
2016 Chile National Olympiad, 6
Let $P_1$ and $P_2$ be two non-parallel planes in space, and $A$ a point that does not It is in none of them. For each point $X$, let $X_1$ denote its reflection with respect to $P_1$, and $X_2$ its reflection with respect to $P_2$. Determine the locus of points $X$ for the which $X_1, X_2$ and $A$ are collinear.
1995 China Team Selection Test, 2
Given a fixed acute angle $\theta$ and a pair of internally tangent circles, let the line $l$ which passes through the point of tangency, $A$, cut the larger circle again at $B$ ($l$ does not pass through the centers of the circles). Let $M$ be a point on the major arc $AB$ of the larger circle, $N$ the point where $AM$ intersects the smaller circle, and $P$ the point on ray $MB$ such that $\angle MPN = \theta$. Find the locus of $P$ as $M$ moves on major arc $AB$ of the larger circle.
1978 IMO Longlists, 46
We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.
1965 IMO Shortlist, 5
Consider $\triangle OAB$ with acute angle $AOB$. Thorugh a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over
a) the side $AB$;
b) the interior of $\triangle OAB$.
1966 IMO Longlists, 17
Let $ABCD$ and $A^{\prime }B^{\prime}C^{\prime }D^{\prime }$ be two arbitrary parallelograms in the space, and let $M,$ $N,$ $P,$ $Q$ be points dividing the segments $AA^{\prime },$ $BB^{\prime },$ $CC^{\prime },$ $DD^{\prime }$ in equal ratios.
[b]a.)[/b] Prove that the quadrilateral $MNPQ$ is a parallelogram.
[b]b.)[/b] What is the locus of the center of the parallelogram $MNPQ,$ when the point $M$ moves on the segment $AA^{\prime }$ ?
(Consecutive vertices of the parallelograms are labelled in alphabetical order.
1967 IMO Longlists, 9
Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$
1961 IMO Shortlist, 6
Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A',B',C'$. Let $L,M,N$ be the midpoints of segments $AA', BB', CC'$; Let $G$ be the centroid of the triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L,M,N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\epsilon$?
1992 IMO Longlists, 17
In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.
1963 IMO, 2
Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting segment $BC$.
2017 Philippine MO, 4
Circles \(\mathcal{C}_1\) and \(\mathcal{C}_2\) with centers at \(C_1\) and \(C_2\) respectively, intersect at two points \(A\) and \(B\). Points \(P\) and \(Q\) are varying points on \(\mathcal{C}_1\) and \(\mathcal{C}_2\), respectively, such that \(P\), \(Q\) and \(B\) are collinear and \(B\) is always between \(P\) and \(Q\). Let lines \(PC_1\) and \(QC_2\) intersect at \(R\), let \(I\) be the incenter of \(\Delta PQR\), and let \(S\) be the circumcenter of \(\Delta PIQ\). Show that as \(P\) and \(Q\) vary, \(S\) traces the arc of a circle whose center is concyclic with \(A\), \(C_1\) and \(C_2\).
1973 IMO Shortlist, 5
A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles.
1986 Traian Lălescu, 2.2
Let be a line $ d: 3x+4y-5=0 $ on a Cartesian plane. We mark with $ \mathcal{L} $ de locus of the planar points $ P $ such that the distance from $ P $ to $ d $ is double the distance from $ P $ to the origin. Let be $ B_{\lambda } ,C_{\lambda }\in\mathcal{L} $ such that $ C_{\lambda } -B_{\lambda } +\lambda =0. $ Find the locus of the middlepoints of the segments $ B_{\lambda }C_{\lambda }, $ if $ \lambda\in\mathbb{R} $ is variable.
1969 IMO Longlists, 1
$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$
1986 IMO Shortlist, 1
Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.
1978 IMO, 2
We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.
1971 IMO Shortlist, 4
We are given two mutually tangent circles in the plane, with radii $r_1, r_2$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of $r_1$ and $r_2$ and the condition for the solvability of the problem.
2007 Sharygin Geometry Olympiad, 18
Determine the locus of vertices of triangles which have prescribed orthocenter and center of circumcircle.
1966 IMO Longlists, 28
In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of
[b]a.)[/b] all vertices $A$ of such triangles;
[b]b.)[/b] all vertices $B$ of such triangles;
[b]c.)[/b] all vertices $C$ of such triangles.
1971 IMO Longlists, 17
We are given two mutually tangent circles in the plane, with radii $r_1, r_2$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of $r_1$ and $r_2$ and the condition for the solvability of the problem.
2017 Ukrainian Geometry Olympiad, 3
Circles ${w}_{1},{w}_{2}$ intersect at points ${{A}_{1}} $ and ${{A}_{2}} $. Let $B$ be an arbitrary point on the circle ${{w}_{1}}$, and line $B{{A}_{2}}$ intersects circle ${{w}_{2}}$ at point $C$. Let $H$ be the orthocenter of $\Delta B{{A}_{1}}C$. Prove that for arbitrary choice of point $B$, the point $H$ lies on a certain fixed circle.
1986 IMO Longlists, 47
Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.
1969 IMO Longlists, 53
$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$
1965 IMO, 5
Consider $\triangle OAB$ with acute angle $AOB$. Thorugh a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over
a) the side $AB$;
b) the interior of $\triangle OAB$.
2006 Sharygin Geometry Olympiad, 14
Given a circle and a fixed point $P$ not lying on it. Find the geometrical locus of the orthocenters of the triangles $ABP$, where $AB$ is the diameter of the circle.
1969 IMO Shortlist, 12
$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.